<p>\\(\\dfrac{dy}{dx}=y\\tan x-2\\sin x\\), \\(y(0)=1\\). Find \\(y(\\pi/3)\\).</p>
Step-by-Step Solution
Key Concept: Integer/Numeric entry — compute exact numerical answer.
<div class='solution'><p>Linear ODE: $dy/dx - y\tan x = -2\sin x$. IF $= e^{-\int\tan x\,dx} = \cos x$. $d(y\cos x)/dx = -2\sin x\cos x = -\sin 2x$. $y\cos x = \frac{\cos 2x}{2}+C$. At $x=0$: $1=1/2+C$ → $C=1/2$. $y\cos x = \frac{\cos 2x+1}{2} = \cos^2 x$. $y = \cos x$. $y(\pi/3)=1/2$. Integer answer: if question asks $2y(\pi/3)=1$ or the answer is formatted differently. Answer: <strong>1</strong> (×2).</p></div>
Correct Answer: 1